A144006 -id:A144006 - cp's OEIS frontend

A014621 Triangle of numbers arising from analysis of Levine's sequence A011784.

1, 1, 3, 1, 15, 10, 3, 1, 105, 105, 55, 30, 10, 3, 1, 945, 1260, 910, 630, 350, 168, 76, 30, 10, 3, 1, 10395, 17325, 15750, 12880, 9135, 5789, 3381, 1806, 910, 434, 196, 76, 30, 10, 3, 1, 135135, 270270, 294525, 275275, 228375, 172200, 120960, 78519, 48006, 28336, 16065, 8609, 4461, 2166, 1018, 470, 196, 76, 30, 10, 3, 1

Offset: 1

Colin Mallows

			Triangle begins:
    1;
    1;
    3,    1;
   15,   10,   3,   1;
  105,  105,  55,  30,  10,   3,  1;
  945, 1260, 910, 630, 350, 168, 76, 30, 10, 3, 1;
10395, 17325, 15750, 12880, 9135, 5789, 3381, 1806, 910, 434, 196, 76, 30,
  10, 3, 1;
135135, 270270, 294525, 275275, 228375, 172200, 120960, 78519, 48006, 28336, 16065, 8609, 4461, 2166, 1018, 470, 196, 76, 30, 10, 3, 1;
2027025, 4729725, 5990985, 6276270, 5853925, 4996530, 3999765, 2997225, 2115960, 1432725, 938644, 593646, 364551, 215940, 123639, 68886, 37276, 19485, 9959, 4911, 2301, 1063, 470, 196, 76, 30, 10, 3, 1;

Roland Miyamoto, Rows n = 1..50 of triangle, flattened (rows n = 1..6 from _Colin Mallows_)
Roland Miyamoto, Comments on A014621, Oct 14 2022.
Roland Miyamoto and J. W. Sander, Solving the iterative differential equation -gamma*g' = g^{-1}, in: H. Maier, J. & R. Steuding (eds.), Number Theory in Memory of Eduard Wirsing, Springer, 2023, pp. 223-236, alternative link.
Roland Miyamoto, Python3 program a014621.py implementing below formulae.
Roland Miyamoto, Polynomial parametrisation of the canonical iterates to the solution of -gamma*g' = g^(-1), arXiv:2402.06618 [math.CO], 2024.

Cf. A011784, A014622 (row sums), A144006.

Python
```
# See Miyamoto link.
```

From Roland Miyamoto, Nov 20 2022: (Start)
The n-th row contains 1 + (n-1)*(n-2)/2 numbers a(n,k), where n >= 1 and k = 0..(n-1)*(n-2)/2.
Let f be a solution to the iterative differential equation f(f(x))*f'(x) = -1 defined on some nonnegative interval and let tau=f(tau) be a fixed point of f. Then the n-th derivative of f at tau is
  f^{(n)}(tau) = Sum_{k=0..(n-1)*(n-2)/2} (-1)^(n+k)*a(n,k)*tau^(2-3*n-k).
Thus, a(n,k) can be calculated recursively using the equations
  0 = (f ° f * f')^{(n)} = Sum_{k=0..n} binomial(n,k) (f ° f)^{(n-k)}*f^{(k+1)} for n=1,2,... (End)

More terms from Roland Miyamoto, Nov 20 2022

Offset corrected by Max Alekseyev, Sep 19 2023

A375720 Irregular triangle, read by rows: Coefficients of the polynomials P_n, n>=2 such that the series f(x) = c + c(x-c) + Sum_{n>=2} P_n(c)/c^((n-1)*(n+2)/2+1) (x-c)^n/n! satisfies f(c) = c and f'(f(x)) = x near the fixed point c in (0,oo).

Original entry on oeis.org

1, -1, 1, 3, -1, -3, -10, -15, 1, 3, 10, 30, 55, 105, 105, -1, -3, -10, -30, -76, -168, -350, -630, -910, -1260, -945, 1, 3, 10, 30, 76, 196, 434, 910, 1806, 3381, 5789, 9135, 12880, 15750, 17325, 10395, -1, -3, -10, -30, -76, -196, -470, -1018, -2166, -4461, -8609, -16065, -28336, -48006, -78519, -120960, -172200, -228375, -275275, -294525, -270270, -135135

Offset: 2

Lucas Larsen, Aug 26 2024

The indices in each row range from 0 to (n-3)*(n-2)/2

When c = phi = (1+sqrt(5))/2 the series becomes the Taylor expansion of f(x) = phi^(-1/phi)*x^phi centered at phi, in particular the radius of convergence is positive for at least this choice of c.

			Triangle begins:
  1;
  -1;
  1, 3;
  -1, -3, -10, -15;
  1, 3, 10, 30, 55, 105, 105;
  -1, -3, -10, -30, -76, -168, -350, -630, -910, -1260, -945;
  ...
Polynomials are:
  P_2(c) = 1
  P_3(c) = -1
  P_4(c) = 1 + 3c
  P_5(c) = -1 - 3c - 10c^2 - 15c^3
  etc.
Hence the series begins
f(x) = c + c*(x-c) + c^(-1)(x-c)^2/2 - c^(-4)(x-c)^3/6 + (3c^(-7) + c^(-8))(x-c)^4/24 + ...

Cf. A144006.

def T(n,k):
    c = {(-1,):1} #Polynomial in infinitely many variables (function iterates)
    for _ in range(n-2):
        cnext = {}
        for key, value in c.items():
            key += (0,)
            for i, ni in enumerate(key):
                term = tuple(nj-2 if j==i else nj-1 if j<=i+1 else nj
                             for j,nj in enumerate(key))
                cnext[term] = cnext.get(term,0) + value*ni
                if cnext[term] == 0:
                    del cnext[term]
        c = cnext
    pairs = {} #Reduction to single variable (evaluation at fixpoint)
    for key, value in c.items():
        s = sum(key)
        pairs[s] = pairs.get(s,0) + value
    return pairs.get(1+k-(n-1)*(n+2)//2,0)

cp's OEIS Frontend

A144007 Limit of unsigned reversed rows of triangle A144006.

Views

Author

Keywords

Crossrefs

A014621 Triangle of numbers arising from analysis of Levine's sequence A011784.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Python

Formula

Extensions

A375720 Irregular triangle, read by rows: Coefficients of the polynomials P_n, n>=2 such that the series f(x) = c + c(x-c) + Sum_{n>=2} P_n(c)/c^((n-1)*(n+2)/2+1) (x-c)^n/n! satisfies f(c) = c and f'(f(x)) = x near the fixed point c in (0,oo).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Python